Can we keyframe hair particle brownian motion?
Yes that is a bug. When the type is set to hair most keyed settings are snapping back to the keyed value which prevents new keyframes being added or adjusted from the properties panels. This is not hair only settings, physics, field weights and force field weights are also affected, while render settings do work properly.

For a workaround you can add keyframes and adjust settings in the graph editor once a value has a keyframe.